Properties

Label 671.2.bm.a
Level $671$
Weight $2$
Character orbit 671.bm
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [671,2,Mod(47,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([24, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bm (of order \(15\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 3 q^{2} - 12 q^{3} + 55 q^{4} - q^{5} + q^{6} - 6 q^{7} - 20 q^{8} - 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 3 q^{2} - 12 q^{3} + 55 q^{4} - q^{5} + q^{6} - 6 q^{7} - 20 q^{8} - 124 q^{9} - 8 q^{10} - 18 q^{11} - 2 q^{12} - 7 q^{13} - 7 q^{14} + 3 q^{15} + 43 q^{16} - 12 q^{17} + 7 q^{18} - 11 q^{19} - 32 q^{20} - 14 q^{21} - 2 q^{22} - 56 q^{23} - 24 q^{24} + 51 q^{25} + 7 q^{26} - 36 q^{27} - 52 q^{28} - 18 q^{29} - 84 q^{30} + 5 q^{31} + 2 q^{32} + 20 q^{33} - 9 q^{35} + 86 q^{36} - 36 q^{37} + 22 q^{38} - 42 q^{39} + 35 q^{40} + 38 q^{41} - 26 q^{42} - 26 q^{43} - 11 q^{44} + 44 q^{45} - 33 q^{46} + 11 q^{47} + 69 q^{48} + 38 q^{49} - 70 q^{50} + 37 q^{51} + 74 q^{52} - 60 q^{53} - 94 q^{54} - 25 q^{55} + 6 q^{56} + 19 q^{57} + 8 q^{58} - q^{59} - 30 q^{60} + 9 q^{61} + 230 q^{62} + 63 q^{63} - 84 q^{64} + 8 q^{65} + 22 q^{66} - 56 q^{67} - 72 q^{68} - 124 q^{69} - 46 q^{70} + 13 q^{71} + 214 q^{72} + 18 q^{73} - 56 q^{74} + 29 q^{75} + 124 q^{76} - 74 q^{77} - 4 q^{78} + 5 q^{79} - 214 q^{80} - 124 q^{81} - 25 q^{82} - 77 q^{83} - 46 q^{84} + 62 q^{85} + 67 q^{86} - 38 q^{87} + 52 q^{88} + 144 q^{89} - 40 q^{90} - q^{91} + 67 q^{92} - 5 q^{93} - 36 q^{94} - 50 q^{95} - 144 q^{96} + 37 q^{97} + 128 q^{98} - 226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −2.47281 1.10097i 0.911643 + 2.80575i 3.56441 + 3.95867i 0.118847 1.13076i 0.834714 7.94178i −0.206171 + 0.0438230i −2.78282 8.56465i −4.61409 + 3.35233i −1.53881 + 2.66530i
47.2 −2.46938 1.09944i −0.161350 0.496583i 3.55081 + 3.94357i −0.0870451 + 0.828178i −0.147529 + 1.40365i 3.22548 0.685597i −2.76198 8.50051i 2.20649 1.60311i 1.12548 1.94939i
47.3 −2.46644 1.09813i 0.427841 + 1.31676i 3.53917 + 3.93065i −0.393604 + 3.74489i 0.390728 3.71753i −4.24199 + 0.901662i −2.74419 8.44575i 0.876244 0.636628i 5.08318 8.80432i
47.4 −2.46005 1.09528i −0.858220 2.64133i 3.51393 + 3.90262i −0.0532761 + 0.506889i −0.781744 + 7.43780i −1.23013 + 0.261473i −2.70569 8.32727i −3.81303 + 2.77033i 0.686249 1.18862i
47.5 −2.23412 0.994694i 0.0984508 + 0.303000i 2.66361 + 2.95824i 0.0562497 0.535181i 0.0814419 0.774868i 2.72396 0.578996i −1.49685 4.60683i 2.34493 1.70369i −0.658010 + 1.13971i
47.6 −2.15087 0.957629i −0.755329 2.32466i 2.37093 + 2.63318i 0.190967 1.81693i −0.601551 + 5.72338i −2.24928 + 0.478099i −1.12284 3.45573i −2.40649 + 1.74842i −2.15069 + 3.72510i
47.7 −2.06087 0.917558i −0.708558 2.18072i 2.06701 + 2.29564i −0.372272 + 3.54193i −0.540690 + 5.14432i 0.890549 0.189292i −0.759220 2.33664i −1.82643 + 1.32698i 4.01713 6.95788i
47.8 −1.96443 0.874619i 0.796648 + 2.45183i 1.75575 + 1.94996i 0.0324014 0.308279i 0.579461 5.51320i 1.16764 0.248189i −0.414593 1.27599i −2.94977 + 2.14314i −0.333277 + 0.577252i
47.9 −1.96204 0.873555i −0.00351773 0.0108264i 1.74823 + 1.94160i 0.127016 1.20848i −0.00255559 + 0.0243148i −3.78448 + 0.804416i −0.406628 1.25147i 2.42695 1.76328i −1.30488 + 2.26012i
47.10 −1.87859 0.836403i 0.696822 + 2.14460i 1.49128 + 1.65623i 0.203556 1.93671i 0.484704 4.61165i −2.97503 + 0.632363i −0.145313 0.447227i −1.68669 + 1.22545i −2.00227 + 3.46803i
47.11 −1.87605 0.835272i −0.661083 2.03461i 1.48363 + 1.64773i 0.403601 3.84001i −0.459222 + 4.36921i 0.534911 0.113699i −0.137863 0.424298i −1.27554 + 0.926732i −3.96463 + 6.86693i
47.12 −1.80828 0.805099i 0.222654 + 0.685260i 1.28344 + 1.42540i −0.317759 + 3.02328i 0.149080 1.41840i −0.411179 + 0.0873988i 0.0501149 + 0.154238i 2.00705 1.45820i 3.00864 5.21111i
47.13 −1.77387 0.789778i −0.397407 1.22309i 1.18460 + 1.31563i 0.0674340 0.641592i −0.261023 + 2.48347i 2.84339 0.604382i 0.137790 + 0.424075i 1.08903 0.791226i −0.626334 + 1.08484i
47.14 −1.70810 0.760495i 0.501967 + 1.54490i 1.00099 + 1.11171i 0.450831 4.28937i 0.317476 3.02058i 4.54281 0.965603i 0.291227 + 0.896305i 0.292321 0.212383i −4.03211 + 6.98382i
47.15 −1.48297 0.660260i 0.874496 + 2.69142i 0.424988 + 0.471997i −0.286644 + 2.72723i 0.480188 4.56868i 1.00163 0.212902i 0.684658 + 2.10716i −4.05196 + 2.94392i 2.22576 3.85513i
47.16 −1.43570 0.639214i −0.970398 2.98658i 0.314374 + 0.349148i −0.114787 + 1.09213i −0.515864 + 4.90812i 3.78205 0.803899i 0.743116 + 2.28708i −5.55093 + 4.03298i 0.862905 1.49460i
47.17 −1.34961 0.600883i −0.579021 1.78204i 0.122112 + 0.135620i −0.163893 + 1.55934i −0.289350 + 2.75298i −4.75277 + 1.01023i 0.829726 + 2.55363i −0.413364 + 0.300327i 1.15817 2.00601i
47.18 −1.32721 0.590914i −0.369514 1.13725i 0.0740567 + 0.0822483i −0.328701 + 3.12738i −0.181591 + 1.72772i −0.648672 + 0.137879i 0.848203 + 2.61050i 1.27026 0.922898i 2.28427 3.95647i
47.19 −1.08348 0.482395i 0.207448 + 0.638460i −0.397045 0.440964i 0.0813463 0.773959i 0.0832245 0.791828i −1.78786 + 0.380021i 0.950467 + 2.92524i 2.06245 1.49846i −0.461490 + 0.799325i
47.20 −1.04425 0.464931i 0.562144 + 1.73010i −0.463958 0.515278i 0.00867475 0.0825347i 0.217358 2.06802i 2.17979 0.463328i 0.951381 + 2.92805i −0.250198 + 0.181779i −0.0474316 + 0.0821539i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
61.c even 3 1 inner
671.bm even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bm.a 480
11.c even 5 1 inner 671.2.bm.a 480
61.c even 3 1 inner 671.2.bm.a 480
671.bm even 15 1 inner 671.2.bm.a 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bm.a 480 1.a even 1 1 trivial
671.2.bm.a 480 11.c even 5 1 inner
671.2.bm.a 480 61.c even 3 1 inner
671.2.bm.a 480 671.bm even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).